In many applications of quasigroups isotopies and homotopies are more important than isomorphisms and homomorphisms. In this paper, the way homotopies may arise in the context of categorical quasigroup model theory is investigated. In this context, the algebraic structures are specified by diagram-based logics, such as sketches, and categories of models become functor categories. An idea, pioneered by Gvaramiya and Plotkin, is used to give a construction of a model category naturally equivalent to the category of quasigroups with homotopies between them.
Keywords: sketches, finite product sketches, sketch models, quasigroups, homotopies.
2000 MSC: 20N05, 18B99, 18A10
Theory and Applications of Categories, Vol. 11, 2003,
No. 1, pp 1-14.